The Magellanic Stream is a 100deg×10deg filament of gas that lies within the Galactic halo and contains ~2×108 Msolar of neutral hydrogen. In this paper we present data from the H I Parkes All Sky Survey (HIPASS) in the first complete survey of the entire Magellanic Stream and its surroundings. We also present a summary of the reprocessing techniques used to recover large-scale structure in the Stream. The substantial improvement in spatial resolution and angular coverage compared to previous surveys reveals a variety of prominent features, including bifurcation along the main Stream filament; dense, isolated clouds that follow the entire length of the Stream; head-tail structures; and a complex filamentary web at the head of the Stream where gas is being freshly stripped away from the Small Magellanic Cloud and the Bridge. Debris that appears to be of Magellanic origin extends out to 20° from the main Stream filaments. The Magellanic Stream has a velocity gradient of 700 km s-1 from the Clouds to the tail of the Stream, ~390 km s-1 greater than that due to Galactic rotation alone, therefore implying a noncircular orbit. The dual filaments comprising the Stream are likely to be relics from gas stripped separately from the Magellanic Bridge and the SMC. This implies that (1) the Bridge is somewhat older than conventionally assumed; and (2) the Clouds have been bound together for at least one or two orbits. The transverse velocity gradient of the Stream also appears to support long-term binary motion of the Clouds. A significant number of the most elongated cataloged Stream clouds (containing ~1% of the Stream mass) have position angles aligned along the Stream. This suggests the presence of shearing motions within the Stream, arising from tidal forces or interaction with the tenuous Galactic halo. As previously noted, clouds within one region of the Stream, along the sight line to the less distant half (southern half on the sky) of the Sculptor Group, show anomalous properties. There are more clouds along this sight line than any other part of the Stream, and their velocity distribution significantly deviates from the gradient along the Stream. We argue that this deviation could be due to a combination of halo material, and not to distant Sculptor clouds, based on a spatial and kinematic comparison between the Sculptor Group galaxies and the anomalous clouds and the lack of cloud detection in the northern half of the group. This result has significant implications for the hypothesis that there might exist distant, massive high-velocity clouds within the Local Group. Cataloged clouds within the Magellanic Stream do not have a preferred scale size. Their mass spectrum f(MHI)~M-2.0HI and column density spectrum f(NHI)~N-2.8HI are steep compared with Lyα absorbers and galaxies, and similar to the anomalous clouds along the Sculptor Group sight line.